Re: Nikiforoff Temperament summary - 10/15/0910:37 PM

Thank you Jim.

For the tuning of the first step, I guess it is a variant of the tuning of F3-A3-C#4-F4-A4 contiguous major thirds used in many sequences: Randy Potter, Jim Colemann, Sanderson-Baldassin, Bill Bremmer, and many others. So it will be relatively easy to find a way to tune the contiguous major thirds from C#3 to A4 in the order given here.

Re: Nikiforoff Temperament summary - 10/16/0905:52 AM

Re: Nikiforoff Temperament summary - 10/16/0906:56 AM

Gadzar’s question is excellent. It relates to the problem of the relationship between octaves and SBIs for the piano’s scaling. When any one is changed the others have to, also. And this problem is rarely seen as a factor when favoring CM3 tuning, but always as a criticism of fourths and fifths tuning. I have decided that with aural tuning, tuning the best possible ET always requires making estimates and then refining the tuning, regardless of the sequence. But by choosing the tuning sequence, we also choose where these estimates are made and how we deal with them.

Re: Nikiforoff Temperament summary - 10/16/0911:24 AM

Reading Gadzar and Tooner's posts, I get the impression that you are waiting "with baited breath" that this sequence is a breakthrough in tuning theory, or something, which it isn't at all. The title of the article is "One, Two, Three! A Temperament Sequence that is Easy to Learn." It is firmly on the CM3rds "side" of the debate vs. fourths/fifths sequences, if that's the way you want to look at it, since the approach to octave widths and setting the foundation of the CM3rds ladder is the same.

Originally Posted By: Gadzar

For the tuning of the first step, I guess it is a variant of the tuning of F3-A3-C#4-F4-A4 contiguous major thirds used in many sequences: Randy Potter, Jim Colemann, Sanderson-Baldassin, Bill Bremmer, and many others...

Yes, and I have written up my approach as well, which is a mashup of many of those techniques, particularly including the Stebbins "let the piano tell you" technique. See the first nine unnumbered bullet points here.

Originally Posted By: Gadzar

But for the remaing of the sequence, how do you tune:4th/5th5th/4th...

The short answer in the Nikiforoff article is "P4s are slightly expanded while P5s are slightly contracted." The first six steps after the ladder are straight-forward, I think. Choose an octave in the ladder that is already tuned, then tune the two notes in-between that have 4th/5th and 5th/4th relationships. For the two other notes depending on the P4 and P5 relationship, the notes are tuned are the "same-direction" interval test to notes that have already been tuned. The contguous fourths are tuned equal beating.

Originally Posted By: Gadzar

How do you tune them?

Well, "it depends," and I don't think there's anything particularly new here. After the ladder, I tune 4ths a tad under 1bps, and 5ths slightly narrow, maybe 0.2 bps. Then I immediately switch to refinement using combinations of parallel, contiguous, and other interval checks.

I try to be aware of the "confidence" with which I've tuned notes: at the beginning of refinement, the ladder notes have higher confidence than the ones I've just tuned with 4ths and 5ths, and so that weights my decisions as I refine and decide which notes to move. Occasionally I will adjust my ladder notes during refinement if it seems the best way to make the smoothest beat progressions.

Originally Posted By: UnrightTooner

Gadzar’s question is excellent. It relates to the problem of the relationship between octaves and SBIs for the piano’s scaling. When any one is changed the others have to, also. And this problem is rarely seen as a factor when favoring CM3 tuning, but always as a criticism of fourths and fifths tuning.

I agree. The Nikiforoff article does not "solve" this problem, it takes another cut at a good combination of these techniques, arranged to favor novices who are just becoming acquainted with what it means to remember a sequence.

Originally Posted By: UnrightTooner

I have decided that with aural tuning, tuning the best possible ET always requires making estimates and then refining the tuning, regardless of the sequence. But by choosing the tuning sequence, we also choose where these estimates are made and how we deal with them.

What I like about this sequence, is the insight that by extending the ladder down one more "rung," it discovers a means to stay within the CM3rds approach, yet rapidly rough-in the rest of the notes so I can get down to refinement.

I've written before that I'm still faster with my old sequence, but I've been practicing this one, trying to get to the point where I'm fluent. Then maybe I'll have some better thoughts about how it compares.

Re: Nikiforoff Temperament summary - 10/16/0911:32 AM

Originally Posted By: UnrightTooner

It relates to the problem of the relationship between octaves and SBIs for the piano’s scaling. When any one is changed the others have to, also. And this problem is rarely seen as a factor when favoring CM3 tuning, but always as a criticism of fourths and fifths tuning.

Oh, and I forgot I wanted to make one more comment about this -- I think I would disagree with the italicized portion of your statement.

Go all the way back to the original Sanderson article, and I think you'll find that the two-octave temperament with a foundation in CM3rds is attempting to address that issue directly. You set the wide intervals first, compromising the octave widths as necessary to get good single and double beat rates, and then proceed from there.

Maybe someone like Bernhard Stopper would not agree that this is the best way, but I don't think you can say the problem is "rarely seen as a factor."

Re: Nikiforoff Temperament summary - 10/16/0911:33 AM

My contiguous fourths (and fifths) are not equal beating. I may not be able to get them to progress chromatically (due to the pinblock), but the contiguous SBIs should be progressive in the temperament. Otherwise, you may not have the fourth below a note beat faster than the fifth below a note.

Re: Nikiforoff Temperament summary - 10/16/0911:47 AM

Originally Posted By: Jim Moy

Originally Posted By: UnrightTooner

It relates to the problem of the relationship between octaves and SBIs for the piano’s scaling. When any one is changed the others have to, also. And this problem is rarely seen as a factor when favoring CM3 tuning, but always as a criticism of fourths and fifths tuning.

Oh, and I forgot I wanted to make one more comment about this -- I think I would disagree with the italicized portion of your statement.

Go all the way back to the original Sanderson article, and I think you'll find that the two-octave temperament with a foundation in CM3rds is attempting to address that issue directly. You set the wide intervals first, compromising the octave widths as necessary to get good single and double beat rates, and then proceed from there.

Maybe someone like Bernhard Stopper would not agree that this is the best way, but I don't think you can say the problem is "rarely seen as a factor."

The issue that I am addressing is not one of varying octave widths, and the need for a changing beat ratio for the ladder, but determining the correct beat rate for the fourths and fifths.

The Baldassin-Sanderson temperament sequence is the only CM3 sequence that I know of that does address it (and it takes nine notes to form the “mini” temperament) by tuning a series of fourths that all beat at the same speed. But when it gets to fourths that are contiguous, I believe they should be progressive and can be tuned that way. Chromatic fourths are much more difficult.

Re: Nikiforoff Temperament summary - 10/16/0904:26 PM

I think than both sequences, i.e. Sanderson-Baldassin's and Jim Moy's, address the setting of the fourths's (and 5ths) width.

I can not say the same of Nikiforoff's sequence as it tunes 4ths and 5ths without embedding 3rds simultaneously. Though, I've not seen the tests he uses to tune the 4ths and 5ths.

In the Sanderson-Baldassin's you first estimate A#3 as a fourth from F3, then M3 F#3-A#3 a bit (how much?) faster than F3-A3, then you estimate B3 as a fourth from F#3. This second fourth is supposed to be equal tempered than F3-A#3 but here the problem is what do we mean by that. Equal beating? A little, very little faster? How much? They are chromatic fourths and the difference IMO is not audible/tunable. Now you tune G#3 as a fourth from C#4. How do you temper this 4th? Supposedly same than F3-A#3. But again, does that mean equal beating? This time fourths are not chromatic, but 3 semitones apart. Maybe this time we can strive for a little faster beat rate!? Again, in my opinion the difference is negligible and untunable directly. Now we tune C4 as a M3 from G#3 a bit (how much?) slower than A3-C#4, then we tune G3 as a fourth from C4, same tempered than G#3-C#4 and again same questions: equal beating? smooth progression of fourths from F3-A#3/F#3-B3/G3-C4/G#3-C#4? Again I think this progression is not directly tunable. And now we finally have our test interval M3 G3-B3 wich must beat at an appropriate rate to have a smooth progression of all chromatic M3s from F3-A3 to A3-C#4, if this is not the case we have to adjust all the fourths and thirds consequently.

In this sequence the width of the fourths is determined by our decision of making them equal beating or gradually progressing from F3-A#3 to A#3-C#4 (if such a degree of refinement is possible).

In the Jim Moy's sequence we tune pure fifths from F3 and A3 giving temporary notes D3 and C4 and from these notes we place G3 to have to contiguous equal beating 4ths or we can tune 2 contiguous 4ths beating with a very little difference, it is up to us to decide the correct tempering of these contiguous fourths.

So in both sequences the width of the fourths is addressed, the tuning of the temperament octave is well solved taking into account iH as we check 8ves, 5ths, 4ths and 3rds.

Now, as I said, in Sanderson-Baldassin's we have tuned chromatic M3s with a bit faster rate, or in Jim Moy's bracketed hole step and chromatic M3rds. And in both sequences we have tuned equal beating or slightly progressing 4ths. These facts can be source of tiny imperfections in temperament. But in the final step, i.e. refine, we are supposed to correct any imperfection in the chromatic progressions. It is in this step where finally we will cope with all errors we may did previously in our sequence.

My personal preference is to tune a uniform progression of M3s and M10s and let the slow beating intervals fall into place. I am not inclined to tune a progression of fourths or fifths or octaves.

Maybe Sanderson-Baldassin's is more long to tune G3 and B3 (four 4ths and 2 M3s) but when it's done you have a nine note minitemperament to spread out to the hole piano. In the Jim Moy's sequence you go to tune G3 more directly (2 pure 5ths and a note forming 2 contiguous 4ths), but at the end you have only one note tuned. The advantage is that you make an enphasis in tuning bracketed CM3s.

Re: Nikiforoff Temperament summary - 10/16/0904:55 PM

The issue that I am addressing is not one of varying octave widths, and the need for a changing beat ratio for the ladder, but determining the correct beat rate for the fourths and fifths.

Yes, an entirely valid point to bring up, though I don't think I've said anything regarding the correct sizing of the slow beating intervals. I've followed some of your discussions with Bill Bremmer, et.al., so I know you understand the approach the CM3 temperaments are taking. Given that, I think you'll agree that the Nikiforoff temperament doesn't speak toward that issue, it is simply another way of roughing in the notes between the ladder notes.

Perhaps because I have not reproduced the article in full, my summary unwittingly focuses on setting 4ths/5ths, in which case that's my fault, not Eric Nikiforoff's.

Getting off the thread topic, in a two-octave temperament, doesn't the selection of octave widths address your concern about the widths of fourths and fifths, though indirectly? The strategy, and major point of the two-octave temperament is to determine early, what the octave width tradeoffs should be given a particular piano's scaling. It follows that once you've done that, the 4ths/5ths widths fall out naturally, so long as you've correctly tempered the contained intervals (admittedly, no small task, which is why we all talk temperament so much). If you don't like what falls out, then on that piano perhaps different octave widths are appropriate.

Originally Posted By: UnrightTooner

The Baldassin-Sanderson temperament sequence is the only CM3 sequence that I know of that does address it (and it takes nine notes to form the “mini” temperament) by tuning a series of fourths that all beat at the same speed.

The fact that it specifies equal width fourths in its intermediate temperament is not salient to the temperament's strategy. That part of the procedure is only a tactic to locate the note "in the middle" of one of the M3rds, splitting it into whole steps. In some descriptions, those fourths are only temporary reference points, to be re-tuned later. But it's a critical step, as anyone who has taken the PTG examination can tell you: perfectly smooth, parallel M3rds in no way guarantees your 4ths and 5ths aren't messed up entirely! (Which is how the piano is detuned before you take the exam.) I actually use a different technique for splitting that M3rd that I like better, which was summarized by Jim Coleman in the "Baldassin, Sanderson, Kimbell, Tremper" sequence.

One of the things I've noticed in different approaches to temperament is the speed with which people want to get to the refinement phase. Some want to just rough-in the first pass and then do multiple refinement passes with much tweaking and checking. Some want to make that CM3 ladder as "perfect" as possible, and take ensuing movements with equal care such that there is not much refinement to be done after the first pass.

To each their own. I think the same goes for whether we tune 4ths/5ths and check 3rds/6ths or vice-versa.

Re: Nikiforoff Temperament summary - 10/16/0905:03 PM

Originally Posted By: Gadzar

In the Sanderson-Baldassin's you first estimate A#3 as a fourth from F3, then M3 F#3-A#3 a bit (how much?) faster than F3-A3, then you estimate B3 as a fourth from F#3. This second fourth is supposed to be equal tempered than F3-A#3 but here the problem is what do we mean by that. Equal beating? A little, very little faster? How much?

When I was learning the Sanderson-Baldassin sequence, I would read conflicting descriptions. Some would say 1 bps. Some equal. Some "the high fourth a wee bit faster."

I tried them all. And the resulting M3rd was never in a position with much more than a few cents accuracy, on average. Sometimes I'd hit it right on and jump up and down, and other times I was four or five cents off and the width of the third was wrong.

I probably just hadn't practiced it enough, but then I discovered the Kimbell-Tremper method of splitting that M3rd, my accuracy went waaay up. Within only a week's practice I was hitting it to within 1 cent consistently. As a result, I never went back to the "stacked fourths on both sides of a major third" method. (Then I discovered how to move the whole temperament down a M3rd and I was even happier.)

Re: Nikiforoff Temperament summary - 10/16/0905:29 PM

...but then I discovered the Kimbell-Tremper method of splitting that M3rd, my accuracy went waaay up. Within only a week's practice I was hitting it to within 1 cent consistently.

Right! Me too, I have had bad times tuning those 4ths and 3rds before geting a good G3-B3 M3.

So, I countinued my search of a more direct, less "try and error" sequence. Tough I've not tried it, the Kimbell-Tremper method seems to go straight-forward.

I am reluctant of what precision can be attained by tuning 2 pure fifths. There is a large margin left where a fifth is heard as "pure". You must try to tune both “pure” fifths exactly the same width.

I mean: to tune C4 from F3 you can tune a little sharp and then go down until it is pure. That will leave the 5th on the wide side of purity. When tuning D3 you must tune form the flat side up to purity in order to have the same width of the previos fifth tuned. Or visceversa: tune C4 from the flat side and D3 from the sharp side.

And you must also hear at the correct coincident partial and make the appropiate tests M6-M10 or m3-M3, in order to tune both fifths the same type: 3:2 or 6:4.

For me 5ths are not reliable enough.

That’s why I am using a sequence in which after tuning the ladder of contiguous M3s from F3 to A4 I tune a ladder of contiguous m3s: F3-G#3-B3-D4-F4-G#4, dividing the octave into 4 contiguous minor 3rds, just in the same way we tune the contiguous major thirds. This sequence is direct, has plenty of tests available from the beggining and there is no guessing and no try and error in it. Though in some pianos the tuning of m3s is not that obvious and you must use other intervals (which are often available) to solve some notes.

And this is before he started calling the temperament sequence "Baldassin, Sanderson, Kimbell, Tremper." Maybe mercifully so

Originally Posted By: Gadzar

I am reluctant of what precision can be attained by tuning 2 pure fifths. There is a large margin left where a fifth is heard as "pure".

I agree, if you're just listening to the 3:2 directly. But I think I've had good accuracy with the M6-M10th check. Maybe I should go measure exactly how closely I'm getting though. Time for Tunelab.

Originally Posted By: Gadzar

That’s why I am using a sequence in which after tuning the ladder of contiguous M3s from F3 to A4 I tune a ladder of contiguous m3s: F3-G#3-B3-D4-F4-G#4, dividing the octave into 4 contiguous minor 3rds, just in the same way we tune the contiguous major thirds.

Well, are you going to make us beg? Post it! I posted mine, so you post yours (But in a separate thread, please...)

Re: Nikiforoff Temperament summary - 10/16/0905:52 PM

I'm going to quote the money paragraphs from Jim Coleman's article here, just so it shows up on searches:

Originally Posted By: Jim Coleman: SAT- learning aural tuning

The standard method of locating the B3 is to tune two contiguous 4ths upfrom F3 and two contiguous 4ths down from F4. If these are carefully doneso that each beats exactly at 1 bps, then one can balance the B3 betweenthe G3 and D#4 so that they, as contiguous 3rds will be in a 4 to 5 ratio,just as the F3 and A3 3rds are in a 4 to 5 ratio. The speeds may not be thesame in relation to the previously tuned 3rds, but in respect to each otherthey must be in a 4 to 5 ratio. For beginners, this may be a little diffi-cult. And besides, it will be necessary to redo each of the 4 4ths.

A simpler method is to do as Michael Kimbel and Fred Tremper have suggestedto me. Tune 2 perfect 5ths, one down from C#4 to F#3 and the other up fromA3 to E4, then tune B3 as balanced between F#3 and E4. One of the object-ions to this has been that the 2nd and 3rd partials of notes in this areaare often a little irregular and may give unpredictable results becausethese are the partials which are used in 5ths. Never-the-less, it ispossible to tune these intervals with more precision and you have half asmany possibilities to go wrong as with tuning 4 4ths, and then trying toestimate the 4 to 5 ratio with decimal point accuracy.

Re: Nikiforoff Temperament summary - 10/16/0908:27 PM

I find very constuctive and interesting to read and try others efforts and sequences to tune E.T. Even if I'll have to re-start from zero and quit what I was doing until now.

The alternative is to stick to a sequence, and remain sticked to it forever. We still see tuners that use the more than 100 years old Braid-White sequence or a variant of it and are affraid to switch to a more modern, actual, accurate and reliable method, like the ones we are talking about here.

If I ever find a better sequence than the one I use now, I will switch volontiers!

Re: Nikiforoff Temperament summary - 10/18/0901:57 AM

After trying this, I can see the advantage of having the Cm3 ladder in parallel to the CM3 one. With the sequence I have been using for a while now, and posted earlier, even a small mistake in the "two-5ths to make contiguous 4ths procedure" can cause the "other" CM3rds ladder that is a whole step off from the first one to be "shifted" in the wrong direction, even though it progresses well chromatically. Same thing goes for the original Baldassin-Sanderson technique. For that reason, I have always looked to the earliest 4th/5th check I can make in order to make corrections.

But now after some reflection on the Nikiforoff temperament sequence, I am re-considering something I read in Kent Swafford's Every Which Way Temperament:

Originally Posted By: Gregory Graham by way of Kent Swafford

If one is looking for the one-pass, bullet-proof, set-it-in-stone temperament sequence, then one is on a wild goose chase, wasting time, and asking the wrong question.

The simplicity of the Nikiforoff sequence gets me to the refinement steps faster, while still maintaining the robustness of the CM3rds approach. I think the combination of the two is very powerful. So it has caused me to refocus some of my study efforts on making my refinement techniques more efficient, rather than concentrating so hard on the temperament sequence.

The Cm3 technique appears to be very powerful, but it also looks like a lot of practice to make it efficient, and even then I wonder if it can be done "mindlessly."

Re: Nikiforoff Temperament summary - 10/19/0907:37 AM

Jim:

Excuse me if this has been mentioned before. After a set of CM3s is tuned, and then the fourths and/or fifths are tuned to the starting note, there is a chromatic pair of either M6s or m3s that can be used to check the accuracy of the ladder and/or the fourths and fifths.

Example: F3 A3 C#4 are tuned as part of a ladder. E3 and D4 are tuned to A3. The beat rate of M6 E3-C#4 can be compared to M6 F3-D4. If the E3 and D4 are tuned beatless this can be used solely as a way to refine the ladder of CM3s. And if tempered, can be used as a way to refine the fourths. When these test are continued for all the fourths and fifths for the F3’s and C#’s., there is a nine note mini-temperament that has sufficient cross checks to ensure that the fourths and fifths have the proper beatrate. I wonder if there is a number theory significance to the nine note mini-temperament. Baldassin-Sanderson produces one, and it is even significant with 4ths and 5ths sequences where a ladder of CM3s is produced with the ninth note that is tuned.

Sorry that I don’t have time to read all the posts carefully, just wanted to toss this in.

Re: Nikiforoff Temperament summary - 10/19/0911:21 AM

...E3 and D4 are tuned to A3. The beat rate of M6 E3-C#4 can be compared to M6 F3-D4.

Yes, I follow this.

Originally Posted By: UnrightTooner

If the E3 and D4 are tuned beatless this can be used solely as a way to refine the ladder of CM3s. And if tempered, can be used as a way to refine the fourths.

Could you explain this in further detail? To what are you comparing E3 and/or D4 if they are both tuned beatless to A3? I follow using the tempered 4ths, but I'm not sure what you are doing with the beatless ones.

Originally Posted By: UnrightTooner

I wonder if there is a number theory significance to the nine note mini-temperament. Baldassin-Sanderson produces one, and it is even significant with 4ths and 5ths sequences where a ladder of CM3s is produced with the ninth note that is tuned.

I don't know. My impression has always been that the size of the 4ths in that sequence was not particularly important, except that they all needed to be the same so that both sides of the the resulting G3-B3 M3rd are positioned symmetrically. If it was not, then you compensated by adjusting your 4ths again. Which was a lot of work since there were four of them. And when you've got G3-B3, then the sequence goes back and re-tunes those 4ths anyway by bracketing M3rds.

Re: Nikiforoff Temperament summary - 10/19/0911:49 AM

Jim:

When using untempered fourths, there is less chance that the beat ratio between the fourths is not correct for the piano and cause an error in the chromatic M6 check of the CM3s.

My understanding of the workings of the Baldassin-Sanderson sequence is that there is only one possible set of frequencies (for a given set of initial CM3s) where the fourths will all beat the same and the other M3s will beat progressive. The beat rate of the fourths can be expected to be different depending on the scaling and the stretch, but for a given iH curve and initial set of CM3s there is only one beat rate for the fourths that will work. Of course how well a particular tuner can do this on a particular piano is where the rubber meets the road!

Re: Nikiforoff Temperament summary - 10/20/0902:08 AM

When using untempered fourths, there is less chance that the beat ratio between the fourths is not correct...

Really?!

Take this example of the contrary:

To set A4 exactly to 440 hz. we do not tune a pure, beatless, unison: A4-fork, but we tune instead an equal beating F2-Fork / F2-A4 M17th, which is a lot more accurate.

Another example is given by the tuning of octaves. We usually approach the octave by tuning it beatless, then refining it with tests by comparing beat rates of certain meaningful intervals, I.E. M3-M10, m3_M6, M3,-M17, etc. Here again beat rates are more accurate than purity.

In general, it is more accurate to tune an interval making it to beat at some given rate by comparison to another interval( i.e. equal beating, faster than..., slower than..., etc.) than making it beatless, because the margin left by this purity or beatless character is really very large.

Re: Nikiforoff Temperament summary - 10/20/0907:29 AM

Gadzar:

I agree that some speeds of beating are easier to detect than others. It also depends on the interval. Nobody suggests tuning the strings of a unison to be equal beating to another interval. Why? I would say because a unison is unambiguous. Unless there is a flaw, all the partials match. Other intervals like the fifth and especially the octave can be ambiguous. They can have more than one beat. The fourth is less likely to have more than one beat because of the faintness of the 8:6 partial match.

But what I am talking about here is using the fourths to produce a pair of chromatic M6s in order to refine the CM3s. It does not matter if the fourths are just or not, but they must be the same width. It is more likely to have them the same width by being beatless. It may not be know what beat ratio they should be to each other, so having them beat could cause an error.

Re: Nikiforoff Temperament summary - 10/20/0902:33 PM

Jim:

If the lower M6 beats the same speed or faster or a lot slower than the upper M6, then there is room for improvement in the CM3s. Progressive RBIs that are 1 semi-tone apart require much closer tuning than RBIs that are 4 semi-tones apart.

Re: Nikiforoff Temperament summary - 10/20/0906:50 PM

Alright, it's a parallel interval progression check, I get it. And it's nice, for several reasons if I'm using it early on in setting my M3 ladder:

I can use an M3-M6 check to see they're beatless, for accuracy.

They involve two notes (E3, D4) that haven't been set yet (if you're using the "let the piano tell you" technique) and the other two notes in the test intervals have already been set. So the two M6ths are "high confidence" intervals.

One of the M6ths is off of C#4, which while setting an F3-F4 CM3 ladder is the first note adjusted that sets a "good" M3rd width.

So if you're having trouble with that "let the piano tell you" technique, this would be one more indicator to check against.

Re: Nikiforoff Temperament summary - 10/21/0907:58 AM

Jim:

I really should mention that I don't use this sequence, just happen to know about the check. I believe that an evenly progressive ladder of CM3s is not appropriate for challenging pianos, which is most of my business.

This same check (sometimes with m3s) can be used for the tempering of the fourths and fifths, once the CM3s are really set, especially when the fourths or fifths are tuned to all notes of the initial CM3.

Re: Nikiforoff Temperament summary - 10/23/0907:17 AM

Originally Posted By: UnrightTooner

Jim:

If the lower M6 beats the same speed or faster or a lot slower than the upper M6, then there is room for improvement in the CM3s. Progressive RBIs that are 1 semi-tone apart require much closer tuning than RBIs that are 4 semi-tones apart.

I really don't see your point. How can you improve CM3 by introducing two more 4ths in the equation? If the beat rates of the M6s are not in a good progression it means nothing about CM3s, you may have misplaced your 4ths! The tests you make to set your fourths are the beat rates of M3s and M6ths; I don’t see why they would be more reliable than the tuning of CM3s. The fact that the M6 are chromatic doesn’t add precision at all. They may be equal beating or in an inverted progression without notice!

Isn't it easier to tune the CM3s progression directly?

The question here is:

Why don't you trust the setting of CM3's, all by itself?

All the tests you can imagine to check the CM3s make use of the same skills, estimations and appreciations needed to set the CM3s, and need the tuning of additional notes, subject to more errors, so how can it be more reliable than the direct estimation of the beat rate of M3s?

I have measured with my ETD and you can hear a 0.3 cents deviation in the tuning of F3 with easy. I’ve posted the procedure and results here in PW. And with the same accuracy you can then tune C#4. The tuning of A3 and F4 is by definition the width you select for the octaves in the temperament region. That gives the required precision to tune the CM3. No more tests or refinement are required. It wouldn’t be perfect, of course, but it would be accurate enough to set a good temperament. Trying to make corrections at this point of the sequence, in the setting of the CM3, is to waste time and efforts. There is the last step in the sequence: refine, which is best suited and placed to strive for perfection.

Once you have tuned all the notes between F3-A4, or even best and more suitable: the hole midrange C3-C5, then you can refine by playing chromatic runs of M3, P4ths, P5ths, M6ths, 8ves, M10ths and P12ths. If you find an uneven interval you can use tests of contiguous 3rds, 4ths and 5ths to detect the note(s) in fault and make corrections.

Re: Nikiforoff Temperament summary - 10/26/0908:06 AM

Gadzar:

When I tune an SBI, I can hear the slightest change in pitch because the beat rate is slow, or in the case of just SBIs, zero. I do not hear this when tuning RBIs.

Lets see, 0.3 cents at the fourth partial of the A3 is about 1 beat every 7 seconds. Do you think it is more likely to hear that sort of difference with an interval that is at or less than 1 bps or one that is at or more than 8 bps? Do think it is more likely to hear this difference between intervals with a ratio of 4:5, as in CM3s or between intervals with a ratio of 15:16 as in chromatic intervals?

But even so, is 0.3 cents enough accuracy to guarantee that all RBIs are progressive? As a proponent of CM3 tuning, you should be able to answer this.

But really, there are so many tests and checks available. Why not just let everyone mention what they prefer, and why? I see no reason to be dogmatic.

Re: Nikiforoff Temperament summary - 10/26/0904:33 PM

Tooner, I think how well each of us can discern changes in beat rates in both SBI's and RBI's has much to do with the time spent listening for them. Coming from a background more in line with a modified Braid White temperament I pretty well found a M3rd in the F4-A4 region almost useless, kind of like a revving chainsaw. Some people find it useful as the upper end of CM3rds and such but perhaps they spent a lot of time working in the +12 bps range. As a chromatic progression I think anything above 12 bps is not reliable....for me at least.

Re: Nikiforoff Temperament summary - 10/27/0907:29 AM

Re: Nikiforoff Temperament summary - 10/27/0904:39 PM

Tooner,

Of course I can answer.

When tuning CM3s you can set these five notes (F3, A3, C#4, F4 and A4) in a way that every major third beats faster than its contiguous lower major third. And you can easily hear a deviation of 0.3 cents in the tuning of F3. Not because you can estimate its beat rate all by itself, but because of the way the progression of CM3 is affected by such a deviation. Remember that when you move F3 you have to readjust F4, so the beat rates of the three M3s: F3-A3, C#4-F4 and F4-A4 are all affected simultaneously and that changes the hole progression in a way that lets you detect the right spot for F3 with an acuracy of 0.3 cents.

That precision in the tuning of the CM3s makes unecesary any further testing. With these five notes tuned with that acuraty you can tune the rest of the temperament octave or tenth, and then you can refine the tuning in order to have an even chromatic progression of all the intervals (M10ths, 8ves, M6ths, P5ths, P4ths, M3rds).

Re: Nikiforoff Temperament summary - 10/27/0904:47 PM

Re: Nikiforoff Temperament summary - 10/28/0908:01 AM

Gadzar:

We can look at this both subjectively and objectively.

Subjectively, you seem to be able to hear a very slight difference in the progression of CM3s. Objectively, a change of +0.3 cents on C#4 of a perfect, theoretical set of CM3s will change the CM3 ratio from 3.97:5 to 4.15:5. That is not very much.

Subjectively, I do not hear small differences in the progression of CM3s. I do hear any slight difference in SBIs, though. And I can hear whether chromatic RBIs are in even progression easier than I can hear CM3s. Objectively, it requires greater accuracy to tune progressive chromatic M3s than it does to tune progressive CM3s.

No, 0.3 cents is not enough accuracy to guarantee progressive M3s and M6s, although progressive M3s and M6s can be tuned with errors greater than 0.3 even up to 0.9 cents. (I think this is part of the reason for the 0.9 cent allowable error on the PTG exam.)

I am pretty sure that you understand math enough to follow this. A M3 is 13.7 cents wide of just. If a sample M3 beats 8 bps at 13.7 cents, it will beat 16 bps at 27.6 cents. Twice as fast for twice as wide, same as if the M3 was an octave higher. This means that the 12th root of two can be used to determine how much a M3 (or any other interval) can be changed before it beats the same speed as the interval chromatically higher. So for the sample M3 to beat the same speed as the next higher M3 (8 bps * 1.059 = 8.472 bps) the interval would need to be widened to 14.5 cents, or 0.8 cents wider than theoretical (13.7 * 1.059 = 14.51 cents). But if we start with perfect theoretical ET and flatten the lower note of a M3 0.8 cents and sharpen the upper note 0.8 cents, then the interval will have a beat speed as fast as the interval two semi-tones higher. A tolerance of +/- 0.4 cents is not accurate enough either, because one M3 could be widened by 0.8 cents and the next narrowed by 0.8 again causing the beat rates to be unprogressive. Although a tolerance of 0.2 cents can allow some chromatic M3s to beat at the same speed, they would not be unprogressive. So I believe to guarantee that all M3s and M6s to be progressive requires a tolerance of +/- 0.2 cents, although +/- 0.4 cents will usually result in progressive M3s and M6s (but I have not used statistical analysis to work out a bell curve.) If you choose, you can work out the allowable tolerance for a progressive set of CM3s in the same way.

The reason I chose M3s instead of M6s for calculating the allowable tolerance is because M6s are 2 cents further from just intonation than M3s, and do not require as close a tolerance for progressiveness. The closer to just intonation, the closer the tolerance that is required. This is an objective reason for using SBIs. Since they are 7 times more just than M3s, when they are out of progression, they show an error seven times smaller than M3s would. In a typical piano, this is a smaller tolerance than the pin and string can be reliably set.

Once I understood all this, including my personal abilities and limitations to hear RBIs and SBIs, the choice for me was obvious. I choose to tune with intervals that have a tolerance greater than I can actually tune. It is like doing carpentry work that requires +/- 1/8 inch tolerances. I want a tape measure with 1/16 inch marks, not ¼ inch marks.

And now you see that the ratio of the latest is 4:3 instead of 6:5-5:4 of the two preceding ratios. That is a very easy to hear difference: F4-A4 beats 4 times for 3 beats of C#4-F4!

In simple words F4-A4 beats too fast and you can distinctly hear the uneveness in the progression!

You need no much training in apreciating the difference between 5:4 and 4:3 ratios. Remember that you are hearing and comparing these ratios, you are not trying to estimate a single 5:4 ratio, you are evaluating the whole progression.

Of course it is not possible to distiguish the change in the beat rate of F3-A3 from 6.93 to 7.08. But after you adjust F4 and you play the four CM3s, you can easily detect the uneveness at F4-A4 which will beat too fast compared to the other CM3s.

In the same way, if F3 was tuned 0.3 cents sharp then you will find that F4-A4 will beat too slow in the progression.

Could you please borrow an ETD, or install the trial version of Tunelab for free in your computer and give it a try?

You will change your mind if you do!

You'll see that your ear is accurate enough to detect that 4:3 to 6:5-5:4 ratio difference and you can tune F3 with an accuraty of 0.3 cents.

About 0.3 cents being enough acuraty, please read my words: 0.3 cents in the tuning of F3 from A3. That means just that. A3 is tuned to A4 and F4 is tuned to F3 with the precision you can achieve in tuning your octaves.

That doesn't mean you can have F3 0.3 cents flat with A3 0.3 cents sharp, that would be a "mathematical" conception of acuraty, based on random errors, which is not the case.

And you seem to ignore consistently that this is only the beggining part of the sequence, after that you will refine the tuning of your temperament and correct notes testing all other kind of intervals.

Re: Nikiforoff Temperament summary - 10/30/0908:38 AM

I posted: “Objectively, a change of +0.3 cents on C#4 of a perfect, theoretical set of CM3s will change the CM3 ratio from 3.97:5 to 4.15:5. That is not very much.”

Objectively, our math agrees: 4.15:5 is the same ratio as 4.98:6.

Subjectively, our opinions differ. I do not think this is much of a difference. And btw 3.97:5 is the same ratio as 4.76:6, not far at all from 5:6 and also the same ratio as 3.18:4 not far at all from 3:4.

Although I used to do things a little differently, I now prefer to use each new note that I tune in order to refine the others already tuned, and not wait until the end for refinement. A fourth and fifth sequence is great for this.

I consider much of this to be in the realm of personal abilities and preferences. I do not feel a need for us to agree. Do you?